# acosh, acoshf, acoshl

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< c‎ | numeric‎ | math

Defined in header ``<math.h>`` |  |
float acoshf( float arg ); | (1) | (since C99)
double acosh( double arg ); | (2) | (since C99)
long double acoshl( long double arg ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define acosh( arg ) | (4) | (since C99)

1-3) Computes the inverse hyperbolic cosine of arg.
4) Type-generic macro: If the argument has type long double, `acoshl` is called. Otherwise, if the argument has integer type or the type double, `acosh` is called. Otherwise, `acoshf` is called. If the argument is complex, then the macro invokes the corresponding complex function (cacoshf, cacosh, cacoshl).

### Parameters

arg | - | floating-point value representing the area of a hyperbolic sector

### Return value

If no errors occur, the inverse hyperbolic cosine of arg (cosh-1
(arg), or arcosh(arg)) on the interval [0, +∞], is returned.

If a domain error occurs, an implementation-defined value is returned (NaN where supported).

### Error handling

Errors are reported as specified in `math_errhandling`.

If the argument is less than 1, a domain error occurs.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

- If the argument is less than 1, FE_INVALID is raised an NaN is returned.
- If the argument is 1, +0 is returned.
- If the argument is +∞, +∞ is returned.
- If the argument is NaN, NaN is returned.

### Notes

Although the C standard names this function "arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "inverse hyperbolic cosine" (used by POSIX) or "area hyperbolic cosine".

### Example

Run this code

    #include <errno.h>
    #include <fenv.h>
    #include <float.h>
    #include <math.h>
    #include <stdio.h>
    // #pragma STDC FENV_ACCESS ON

    int main(void)
    {
        printf("acosh(1) = %f\nacosh(10) = %f\n", acosh(1), acosh(10));
        printf("acosh(DBL_MAX) = %f\nacosh(Inf) = %f\n", acosh(DBL_MAX), acosh(INFINITY));

        // error handling
        errno = 0; feclearexcept(FE_ALL_EXCEPT);
        printf("acosh(0.5) = %f\n", acosh(0.5));
        if (errno == EDOM)
            perror("    errno == EDOM");
        if (fetestexcept(FE_INVALID))
            puts("    FE_INVALID raised");
    }

Possible output:

    acosh(1) = 0.000000
    acosh(10) = 2.993223
    acosh(DBL_MAX) = 710.475860
    acosh(Inf) = inf
    acosh(0.5) = -nan
        errno == EDOM: Numerical argument out of domain
        FE_INVALID raised

### References

- C23 standard (ISO/IEC 9899:2024):

  - 7.12.5.1 The acosh functions (p: TBD)

  - 7.27 Type-generic math <tgmath.h> (p: TBD)

  - F.10.2.1 The acosh functions (p: TBD)

- C17 standard (ISO/IEC 9899:2018):

  - 7.12.5.1 The acosh functions (p: 175)

  - 7.25 Type-generic math <tgmath.h> (p: 272-273)

  - F.10.2.1 The acosh functions (p: 379)

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.5.1 The acosh functions (p: 240)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.2.1 The acosh functions (p: 520)

- C99 standard (ISO/IEC 9899:1999):

  - 7.12.5.1 The acosh functions (p: 221)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - F.9.2.1 The acosh functions (p: 457)

### See also

asinhasinhfasinhl (C99)(C99)(C99) | computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) (function)
atanhatanhfatanhl (C99)(C99)(C99) | computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x)) (function)
coshcoshfcoshl (C99)(C99) | computes hyperbolic cosine (\({\small\cosh{x} }\)cosh(x)) (function)
cacoshcacoshfcacoshl (C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)
C++ documentation for acosh

### External links

Weisstein, Eric W. "Inverse Hyperbolic Cosine." From MathWorld — A Wolfram Web Resource.

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